Re: Vector Subspaces

1) the constant 0-sequence has only finitely many non-zero terms. (this is easy).
2) the sum of any two sequences with only finitely many non-zero terms has only finitely many non-zero terms.

(hint: any sequence with only finitely many non-zero terms has a maximim non-zero term, say . if is another such sequence, it has a maximum non-zero term, say . show that if t = max{k,m}, all for n > t are 0).

3) you must verify that the sequence has only finitely many non-zero terms if does. this is also easy.

Re: Vector Subspaces

1) the constant 0-sequence has only finitely many non-zero terms. (this is easy).
2) the sum of any two sequences with only finitely many non-zero terms has only finitely many non-zero terms.

(hint: any sequence with only finitely many non-zero terms has a maximim non-zero term, say . if is another such sequence, it has a maximum non-zero term, say . show that if t = max{k,m}, all for n > t are 0).

3) you must verify that the sequence has only finitely many non-zero terms if does. this is also easy.

Re: Vector Subspaces

you need to be sure W is not empty. the 0-vector is the traditional choice, because every vector space, no matter how small, must possess an additive identity.

you want to be sure the sum of two sequences who have only finitely many non-zero terms, is also such a sequence (this is called closure under vector addition). if only finitely many are non-zero, that means after a finite number, everything else is 0. here is one such sequence:

Re: Vector Subspaces

the style of Fernando's response is fairly typical...symbols for precision (especially when describing elements, sets, and conditions), and words for smoother readability. if a subset W satisfies all 3 conditions laid out in post #3, it is a subspace. in words, you would say (having proven those conditions): "since W is a non-empty set closed under vector addition and scalar multiplication, it is a subspace of V". there is not, to my knowledge, any special symbol for "is a subspace of".

Fernando's post (#6) is a near-perfect demonstration of closure of vector addition, so that leaves you with 2 more conditions to verify.

Re: Vector Subspaces

From original post:
"Let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entries is a subspace of V"

A sequence of zeros does not have any non-zero entries, so it cannot have finiteley many of them. 0 does not belong to W.